Time Value of Money — Interest Paid on Mortgage

Question

You you bought a house worth $328,000. You paid 25% of the purchase price in cash and arranged a 25 year mortgage with a rate of 4.0% compounded semi-annually for the remaining balance. The mortgage has an amortization period of 25 years. How much interest will you pay in the first 7 years (assuming that the first payment is made at the end of the first month)?

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So far, I have that PV=$328,000 * 0.75=$246,000, r=0.00330589 (using effective rate formula: (1+r)^6=(1+0.04/2) ) and n=25 * 12=300. Using the present value of an ordinary annuity:

PV=PMT[(1-(1+r)^-n)/r]

I solved for PMT and got PMT=$1294.009652 for the monthly payments. The number of payment periods still remaining after 7 years is 18*12=216. The PV of the outstanding balance (FV of 246,000 - FV of 84 PMTs) is $199,539.6457. However, I don't really know what to do after that. The correct answer is $62,236.46 but I don't know how they got that. How do I calculate the interest paid in the first 7 years?

Answer 1

Edit:

Actually, since you already got the Outstandind Principal right after 84 months = $199,539.6457, you knew that:

Total Principal Paid = 246,000 - 199,539.6457 = $46,460.3543

Total Interest Paid = 84 x 1,294.009652 - 46,460.3543 = $62,236.456468

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Start with PMT and r. Your PMT = $1294.009652 and r = 0.00330589 are correct.

Then use second formula at: https://en.wikipedia.org/wiki/Mortgage_calculator#Total_interest_paid_formula

• P = 246,000
• r = 0.00330589
• c = PMT = 1294.009652
• N = 84

You will get $62,236.46

Alternatively just use the Interest paid function of BA II PLUS: https://education.ti.com/download/en/ed-tech/ADF11FB65B284B6195B0A7E9502784BA/5DC3E70F3C8040E499D704B583646E1D/BA_II_PLUS_EN.pdf

You may also play with the relationship:

Total Interest Paid + Total Principal Paid = 84 x 1294.009652

This is a good resource (Starting from 13.3): https://math.libretexts.org/Bookshelves/Applied_Mathematics/Business_Math_(Olivier)/13%3A_Understanding_Amortization_and_its_Applications/13.01%3A_Calculating_Interest_and_Principal_Components

Answer 2

With s as the loan amount

hse = 328000
s = hse (1 - 0.25) = 246000

and r the monthly rate

i = 0.04
r = (1 + i/2)^(2/12) - 1 = 0.00330589

n = 25*12 = 300

the payment amount d is

d = r (1 + 1/((1 + r)^n - 1)) s = 1294.01

The principal balance in month x is given by p(x) (see link)

p(x) = (d + (1 + r)^x (r s - d))/r

and the interest paid in month x is given by int(x)

int(x) = p(x - 1) r
       = d + (1 + r)^(x - 1) (r s - d)

The accumulated interest to month x is given by interestsofar(x)

(formula obtained by induction, summing int(k) from k = 1 to x)

interestsofar(x) = (d - d (1 + r)^x - r s + r (1 + r)^x s + d r x)/r

interestsofar(7*12) = 62236.46